Problem: Two solutions of
\[x^4 - 3x^3 + 5x^2 - 27x - 36 = 0\]are pure imaginary.  Enter these solutions, separated by commas.
Solution: Let $x = ki,$ where $k$ is a real number.  Then the given equation becomes
\[(ki)^4 - 3(ki)^3 + 5(ki)^2 - 27(ki) - 36 = 0,\]which simplifies to
\[k^4 + 3ik^3 - 5k^2 - 27ik - 36 = 0.\]The imaginary part must be 0, so $3ik^3 - 27ik = 3ik(k^2 - 9) = 0.$

Since $k = 0$ does not work, we must have $k = \pm 3$.  Therefore, the pure imaginary solutions are $\boxed{3i,-3i}.$